Rigid Body Issues (Chain is passing through object and not following the mesh)
This example uses the same arrangement mentioned in my comments above in that the falling object is parented to another which is to drag it along (or down) with it.   It's been done in Vers 2.77a. (Blend file below)

With this one, a necklace wouldn't be rigid when it lands, so it's best to keep the array and curve modifier's active so both can be adjusted to emulate real world physics. i.e. deform the necklace's curve.
Soft-body physics etc. can be tried but for this sort of job emulating physics with a few simple keyframes and hooks is arguably more effective and efficient.
To begin with, a circle curve in Edit mode has just 4 control points.  Intermediate CP's can be added by selecting two adjacent points and using &quot;Subdivide&quot;. (press W)  Reshape the necklace as necessary at frame 1.
Six CP's are used here.  Nth, East, Sth &amp; West, with two subdivides, SE &amp; SW.
If it's going to be noticed, the chain's array modifier's link offset-value, will need minute adjustment via a keyframe when the array comes to rest.  This will prevent an overlapping link appearing at the array's original starting link.
Animate the hooks to deform the curve-circle and emulate physics.  The Graph editor makes this a relatively simple task, reshaping each hook's curves accordingly.  (See the Blend file -&gt; Graph Editor)
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The large circle-empty designated &quot;Nav&quot; (navigation), parents everything involved with the necklace. i.e. the original array, the curve circle, and hooks, to keep it all together.  The 'Nav' in turn is parented to the bust.
It's the Nav-empty that's keyframed to take the whole thing down initially, and on impact the hooks take over to deform the necklace's curve-modifier.  The bust when moved, takes the whole thing with it.
You would do the same thing when the bust (or character) is moving about, simply deform the necklace's curve to suit and emulate physics once more.
If you're looking to the physics engine to do it all for you as a game engine might, I'm not sure that would be practical in Blender.